I think, as with most things, it can vary from place to place. In our GP practice, you have a health visitor from the time your baby is two weeks old until the child turns 4. There are set ages that the HV sees all kids (usually when they are due vaccines) but she also runs a weekly drop-in clinic and will see any young child for any reason, even just for like weight checks, etc. After that, things are usually done through the GP. I've never even asked for a well child check. Any time I take my kids to the doctor if they aren't feeling well, they usually check things like vitals etc, and ask if I have any concerns or questions, and that has worked well enough for me. With my daughter (she's 5) the only thing our doctor doesn't usually check is her weight, and I've asked for it and she has done it with no problems.
I will say (and I know this is only my experience) that any concerns I've had have been taken seriously and addressed in a timely manner. Both of my kids had developmental delays from an early age, and at first, it was fairly subtle. Not once did I encounter a person who suggested I worried too much or that I should wait and see. Both of my children have now been diagnosed with autism. My son was just diagnosed in August at age 22 months, which is pretty young, and I put a lot of it down to the health visitor seeing him regularly for the well baby checks and getting the ball rolling for further assessments.